Quantum Geometry and Quantum Field Theory

Transcription

1 Quantum Geometry and Quantum Field Theory Robert Oeckl Downing College Cambridge September 2000 A dissertation submitted for the degree of Doctor of Philosophy at the University of Cambridge

2 Preface This dissertation is based on research done at the Department of Applied Mathematics and Theoretical Physics, University of Cambridge, in the period from October 1997 to August It is original except where reference is made to the work of others. Chapter 4 in its entirety and Chapter 2 partly (as specified therein) represent work done in collaboration with Shahn Majid. All other chapters represent solely my own work. No part of this dissertation nor anything substantially the same has been submitted for any qualification at any other university. Most of the material presented has been published or submitted for publication in the following papers: [Oec99b] R. Oeckl, Classification of differential calculi on U q (b + ), classical limits, and duality, J. Math. Phys. 40 (1999), [MO99] S. Majid and R. Oeckl, Twisting of Quantum Differentials and the Planck Scale Hopf Algebra, Commun. Math. Phys. 205 (1999), [Oec99a] R. Oeckl, Braided Quantum Field Theory, Preprint DAMTP , hepth/ [Oec00b] R. Oeckl, Untwisting noncommutative R d and the equivalence of quantum field theories, Nucl. Phys. B 581 (2000), [Oec00a] R. Oeckl, The Quantum Geometry of Spin and Statistics, Preprint DAMTP , hep-th/ ii

3 Acknowledgements First of all, I would like to thank my supervisor Shahn Majid. Besides helping and encouraging me in my studies, he managed to inspire me with a deep fascination for the subject. Many other people in the department have contributed to making this an enjoyable time for me. In particular, I would like to mention my fellow students Marco Barozzo, Kumaran Damodaran, Tathagata Dasgupta, Steffen Krusch, Hendryk Pfeiffer, Andrew Tolley, and Fabian Wagner. I am grateful to the German Academic Exchange Service (DAAD) and the Engineering and Physical Sciences Research Council (EPSRC) for the funding of much of this work. Finally, my thanks go to my parents, my sister, and my brother, as well as my friends in Cambridge and elsewhere, for their support and care. iii

4 Contents Introduction 1 1 Basics Hopf algebras Quantum Groups Representation Theory (Braided) Monoidal Categories Hopf Algebra Module Categories Quantum Differentials First Order Differential Calculi Exterior Differential Algebras Twisting Theory Hopf Algebras Module Categories Deformation Quantisation Quantum Differentials Covariant Differentials Bicovariant Differentials over Quantum Groups Differential Calculi on U q (b + ) U q (b + ) and its Classical Limits Classification on C q (B + ) and U q (b + ) Classification in the Classical Limit The Dual Classical Limit Remarks on κ-minkowski Space and Integration A Appendix: The Adjoint Coaction on U q (b + ) iv

5 4 Quantum Geometry of the Planck Scale Hopf Algebra The Cocycle Twist Differential Calculi Quantum Poisson Bracket Fourier Theory Spin and Statistics Spin Statistics Braided Categories and Statistics Quantum Groups and Statistics Unifying Spin and Statistics Braided Quantum Field Theory Foundations The Braided Path Integral Gaussian Integration Path Integration Special Cases: Bosons and Fermions Braided Feynman Diagrams Perturbation Theory The Diagrams Bosonic and Fermionic Feynman Rules Braided Quantum Field Theory Special Cases Anyonic Statistics and Quons Symmetric Braided Quantum Field Theory Braided QFT on Quantum Homogeneous Spaces Quantum Homogeneous Spaces Diagrammatic Techniques Braided QFT on Compact Quantum Spaces Braided Spaces of Infinite Dimension Cosemisimplicity and Peter-Weyl Decomposition Quantum Field Theory on Noncommutative R d Noncommutative R d as a Twist A Remark on the Noncommutative Torus Towards Quantum Field Theory v

6 8.3 Equivalences for Quantum Field Theory Perturbative Consequences Additional Symmetries Space-Time Symmetry Gauge Symmetry A Appendix φ 4 -Theory on the Quantum 2-Sphere The Decomposition of SU q (2) and Sq The Free Propagator Interactions Renormalisation A Appendix: Coquasitriangular Structure of SU q (2) Bibliography 126 vi

7 Introduction Reading the title of this dissertation one might ask: What is quantum geometry? or What does it have to do with quantum field theory? The first of these questions we will try to answer immediately. For the second we hope that the following chapters hold at least a partial answer. More familiar perhaps than the term quantum geometry are the terms that it is meant to subsume: noncommutative geometry and quantum groups. Although both belong to the realm of mathematics, their evolution has been very much connected with developments in physics, particularly quantum physics. The story begins with the early days of quantum mechanics. Heisenberg s commutation relations [X, P ] = i imply that the geometry of classical phase space is lost. If coordinates (such as X and P ) on a phase space cease to commute then there can be no such space. Instead, one viewed X and P purely as operators on a Hilbert space. Functional analysis succeeded geometry. It has been dominating quantum mechanics ever since. In spite of this, the idea, in one form or other, that this operator algebra forms some kind of noncommutative geometric space has accompanied quantum mechanics almost from the beginning. However, for a long time, no serious attempt has been made to develop such a generalised geometry. Algebraic geometry is built on a correspondence between spaces and commutative algebras. The correspondence associates with a space the algebra of functions on it. Geometric notions are then expressed in a purely algebraic language. This principle turns out to be the right starting point for a generalised geometry. While for algebraic geometry the spaces are affine schemes, a correspondence that is closer to differential geometry is given by the Gelfand-Naimark theorem. In this case the spaces are topological spaces and the algebras commutative C -algebras. Around 1980, an approach was initiated by Alain Connes which has become widely known as noncommutative geometry [Con80, Con85, Con94]. This approach is very much in the functional analytic tradition of quantum mechanics. It introduces notions of Dirac operator, dif- 1

8 2 Introduction ferential structures, metric structures, vector bundles, etc. in a context of (noncommutative) operator algebras leading to many interesting developments in mathematics. A quite different development originated also around It came out of the study of quantum integrable systems by the (then) Leningrad school of Ludwig D. Faddeev and others. They encountered generalised symmetries that were not described by groups but could be related to groups. From these emerged the q-deformations of Lie algebras and Lie groups [Jim85, Dri85, Wor87, RTF90]. It was then also realised by Vladimir G. Drinfeld that they are examples of a more general structure, called a quantum group. He developed the mathematical theory of quantum groups extensively in the 1980 s [Dri87]. The underlying structure is that of a Hopf algebra, a concept that had appeared already much earlier in the context of group cohomology. To connect the two developments we note that the correspondence between spaces and commutative algebras extends to (algebraic) groups. The extra structure provided by the group multiplication corresponds on the algebra side to the additional structure that makes a commutative algebra a commutative Hopf algebra. Thus, (noncommutative) Hopf algebras are generalisations of groups in the same way as (noncommutative) algebras are generalisations of spaces. The (algebraic) theory that encompasses both is called quantum geometry. 1 Interesting mathematical developments have taken place since the early days. The representation theory of quantum groups is very rich and leads to braided monoidal categories [JS86]. This in turn has led to new insights into knot theory and invariants of 3-manifolds. Many examples of quantum spaces have been investigated since the late 80 s including deformations of R n, of spheres, of projective spaces, etc. Differential structures on quantum groups were introduced [Wor89] as well as quantum principal bundles and connections [BM93]. Besides the purely mathematical attraction of quantum geometry there are very good reasons to expect it to play an important role in physics. This goes far beyond the initial application of quantum groups as symmetries of certain quantum integrable systems. One particularly intriguing idea is that quantum geometry might lend a far better description of space-time at the Planck scale than ordinary differential geometry does. Based on the persistent inability to unite gravity with quantum field theory it has long been conjectured that space-time at short distances might have a discrete or foam-like structure. It seems that, with the emergence of quantum geometry, for the first time the tools for such a description are at hand. As early as the 1940 s it was proposed that space-time coordinates might be noncommuting 1 It is worth mentioning that the term quantum geometry has appeared in recent years also in other contexts, notably in loop quantum gravity and string theory. This is not necessarily related to the meaning we attach to it here.

9 3 observables [Sny47]. This was then motivated by the hope that the infinities arising at very short distances in quantum field theory might be regularised in this way. It took several decades until the emergence of quantum groups and noncommutative geometry for further development of such ideas to occur. Shortly after their introduction, Shahn Majid suggested a role for quantum groups in Planck scale physics as a unified description of the observable algebra and the gravitational curvature [Maj88]. (See Chapter 4 for further developments of this idea.) Around 1990 Alain Connes and collaborators initiated a reformulation of the standard model using noncommutative geometry to describe its internal degrees of freedom [CL90, Con94]. At about the same time quantum deformations of Minkowski space and its symmetries were introduced [PW90, CSSW90, OSWZ92] with the motivation to obtain new physical models by deformation. Braided categories were found to describe generalised particle statistics in the context of algebraic quantum field theory [FRS89, FG90], indicating quantum group symmetries (see Chapter 5). The interest in noncommutative geometry was boosted in 1997 with the emergence of noncommutative spaces in string theory [CDS98, SW99]. (See Chapter 8 for a development on this.) Quantum groups have also appeared in the loop approach at quantum gravity [MS96]. Both indicate an important role for quantum geometry in fundamental physics. The idea of noncommutativity as a regulator has found a successful application in fuzzy physics, where function algebras are approximated by finite dimensional algebras [Mad92, Mad95, GKP96]. This is similar to a lattice approximation, but without the breaking of space-time symmetries. In contrast, the more ambitious aim of a continuous regularisation of quantum field theory by quantum deformation of symmetries (as proposed in [Maj90b]) has been open for some time. It has found its first realisation only recently in the work presented here (see Chapter 9). This dissertation aims at contributing to some of the mentioned developments as well as introducing new approaches. Its most central development is braided quantum field theory, a generalisation of quantum field theory to braided spaces (Chapter 6). This allows the construction of quantum field theories with quantum group symmetries and with general braid statistics. It forms the basis for much of the other material (Chapters 7 9) or is connected to it (Chapter 5). At this point we would like to emphasise that our approach departs considerably from the functional analytic tradition of quantum mechanics (as it seems any framework must do that allows for radical generalisations of the space-time concept). In particular, our path integral no longer corresponds in general to a canonical description with field operators on a Hilbert space. Whether this is a weakness or a strength remains to be seen. The anyonic example of Chapter 7 appears to point towards the latter.

10 4 Introduction Overview Chapter 1 serves as a review for material that underlies much of this dissertation. In particular, some facts about the representation theory of Hopf algebras are recalled which are used in most of the following chapters. Furthermore, aspects of quantum differential calculi are reviewed which are used in Chapters 2 4. Chapter 2 presents an extension of Drinfeld s twisting theory: A twist relating two Hopf algebras gives rise to an equivalence between the module categories of the two Hopf algebras. We show that such an equivalence holds for many other representation categories as well. Furthermore this gives rise to a 1-1 correspondence between differential calculi on quantum spaces related by twist. Differential calculi on the quantum group U q (b + ) are investigated in Chapter 3. We give a complete classification and detailed examination of the classical limit q 1 and its dual. At the end we comment on the relation to κ-minkowski space. The quantum differential geometry of the Planck scale Hopf algebra [Maj88] is investigated in Chapter 4. This is a toy model for physics at the Planck scale. The differential calculi and exterior algebra on it are constructed using results from both previous chapters. The quantum geometry gives rise to a quantum Poisson bracket, suggesting a deviation from standard quantum mechanics at strong curvature. A Fourier transform is developed implementing a T-duality-like self-duality of the model. Chapter 5 examines the quantum group symmetries behind spin and statistics. We show that these symmetries can be unified in the presence of a spin-statistics theorem. The Bose- Fermi case as well as the more general anyonic case are considered. Braided quantum field theory is introduced in Chapter 6. It employs a path integral formalism based on Gaussian integration in braided categories which we review. We obtain a generalisation of Wick s theorem. Based on this we develop perturbation theory leading to braided Feynman diagrams, which generalise ordinary Feynman diagrams. Bosonic and Fermionic path integrals and Feynman rules are recovered as special cases. The remaining chapters concern applications of braided quantum field theory. In Chapter 7 special cases are considered. First, the quons studied by Greenberg [Gre91] are investigated as an example of anyonic statistics. We show that braided quantum field theory provides the path integral counterpart to the usual canonical approach to quons. Then, we show that for symmetric braiding a correspondence between braided and ordinary Feynman diagrams can be established. In particular, this applies to Bosons and Fermions. Finally, methods are developed for quantum field theory on quantum homogeneous spaces and on compact quantum spaces.

11 5 In Chapter 8, quantum field theory on the noncommutative R d arising in a certain limit of string theory is investigated. We show how this space is related to ordinary R d by a Drinfeld twist. Using the twisting theory in conjunction with braided quantum field theory leads to an equivalence relating quantum field theories on ordinary and noncommutative R d. The equivalence exchanges commutativity with noncommutativity and ordinary with a momentum-dependent statistics. We construct φ 4 -theory on the quantum 2-sphere in Chapter 9 using the results about homogeneous and compact quantum spaces of Chapter 7. The free propagator is obtained as well as the tadpole diagram. The latter is responsible for making the ordinary φ 4 -theory divergent. We show that the divergence is regularised in the noncommutative regime q > 1. The formalism suggests that divergences of any degree could be regularised in this way. A diagrammatic interpretation of renormalisation of φ 4 -theory is obtained.

12 Chapter 1 Basics In this chapter we review some basic concepts of quantum group theory that underly much of this work. Section 1.1 recalls definitions of Hopf algebra and (co)quasitriangular structures. This is followed by a brief explanation of our use of the term quantum group. Section 1.2 discusses general aspects of the representation theory of Hopf algebras. Some of this is relevant to all of the following chapters. Section 1.3 introduces the notion of differential calculus and exterior algebra for quantum groups which is needed for Chapters 2, 3, and 4. To fix our notation, we use the symbols, ɛ, and S respectively to denote the coproduct, counit, and antipode of a Hopf algebra, see Section 1.1. For the coproduct we frequently employ Sweedler s component notation h = h (1) h (2) etc., with summation implied. We adopt a similar notation v v (1) v (2) and v v (1) v (2) for left and right coactions, see Section A braiding will generally be denoted by ψ, see Section In this chapter as well as Chapter 2 certain tensor products are denoted by. In the other chapters only the symbol is used. We write a circle for the composition of maps. k denotes a general field. Standard references on Hopf algebras are the books by Sweedler [Swe69] and Abe [Abe80]. For modern accounts of quantum group theory see, e.g., the text-books by Majid [Maj95b], Chari and Pressley [CP94], and Klimyk and Schmüdgen [KS97]. 1.1 Hopf algebras We recall the definitions of a Hopf algebra and those of (co)quasitriangular structures to be found in any text-book on quantum groups. Recall that an associative algebra A is a vector space together with a map : A A A so that ( id) = (id ) as maps A A A A. A unit is an element 1 A so that 1 a = a 1 = a for all a A. Equivalently, a unit is a map η : k A so that 6

13 1.1 Hopf algebras 7 (id η) = (η id) = id as maps A A. Dually, a coassociative coalgebra C is a vector space together with a map : C C C so that ( id) = (id ) as maps C C C C. A counit is a map ɛ : C k so that (ɛ id) = (id ɛ) as maps C C. We also use Sweedler s notation for coproducts c = c (1) c (2), with summation implied. Due to coassociativity, this notation can be extended to multiple applications of the coproduct in the obvious way. A bialgebra B is both an associative algebra with a unit and a coassociative coalgebra with a counit. Furthermore, the two structures are required to be compatible in the obvious way, i.e. (ab) = a (1) b (1) a (2) b (2) and ɛ(ab) = ɛ(a) ɛ(a) and 1 = 1 1 for all a, b B. A Hopf algebra H is a bialgebra with a map S : H H with the property (S h (1) )h (2) = h (1) S h (2) = ɛ(h)1 for all h H. S is called the antipode. Definition A quasitriangular structure (also known as universal R-matrix ) on a Hopf algebra H is an invertible element R H H so that ( id)r = R 13 R 23, (id )R = R 13 R 12, τ h = R( h)r 1 h H, where R with indices denotes the obvious extension to H H H supplemented by the unit and τ denotes the map that exchanges tensor factors. Definition A coquasitriangular structure on a Hopf algebra H is a convolutioninvertible map R : H H k so that R(ab c) = R(a c (1) )R(b c (2) ), R(a bc) = R(a (1) c)r(a (2) b), b (1) a (1) R(a (2) b (2) ) = R(a (1) b (1) )a (2) b (2) for all a, b, c H Quantum Groups We comment here on our use of the term quantum group in this work. Commutative Hopf algebras provide an equivalent description of group structures as in the following well-known example. Example Let G be a compact Lie group. Denote the set of algebraic functions 1 on G by C(G). Then C(G) forms a commutative Hopf algebra as follows: 1(g) = 1, (f h)(g) = f(g)h(g), ɛ f = f(e), f(g, g ) = f(gg ), (S f)(g) = f(g 1 ), 1 These are the functions that are obtained as matrix elements of finite dimensional representations.

14 8 Basics where f, h C(G) while g, g G and e denotes the unit element in G. For the coproduct observe the implicit isomorphism C(G) C(G) = C(G G) which holds due to the Peter-Weyl theorem. This example represents very much the point of view we take on how the concept of a (not necessarily commutative) Hopf algebra generalises the concept of a group. We use the term quantum group with this setting in mind: A Hopf algebra as a generalised function algebra on a group. Consequently, we are usually interested in coactions rather than actions (see Section 1.2.2) and coquasitriangular rather than quasitriangular structures. Due to the self-duality of the axioms of a Hopf algebra there is also the dual point of view, where the Hopf algebra plays the role of the enveloping algebra of a Lie algebra or that of a group algebra. We occasionally employ this point of view as well, in particular in Chapters 3 and 4. An advantage of the function algebra over the enveloping algebra setting lies in the fact that the global structure of the group is not lost. This plays an essential role in Chapter 5. Note that some authors use the term quantum group in a narrower sense, so as to only denote q-deformations of Lie groups and enveloping algebras. 1.2 Representation Theory (Braided) Monoidal Categories Intuitively speaking, a monoidal category is a category with a tensor product. In particular, the category of vector spaces is a monoidal category in this way. The formal definition is as follows: Definition A monoidal category is a category C together with a functor : C C C and a unit object 1 C satisfying the following conditions: (i) There is a natural equivalence Φ U,V,W : U (V W ) (U V ) W satisfying Φ (U V ) (W X) Φ U (V (W X)) id Φ U ((V W ) X) Φ ((U V ) W ) X Φ id (U (V W )) X

15 1.2 Representation Theory 9 (ii) There are natural equivalences ρ U : U 1 U and λ U : 1 U U satisfying U (1 V ) Φ (U 1) V id λ U V ρ id Definition A functor F : C D between monoidal categories is a monoidal functor if F(1 C ) = 1 D and there is a natural equivalence c U,V : F(U) D F(V ) F(U C V ) satisfying the associativity condition (i) c D id F(U C V ) D F(W ) c (F(U) D F(V )) D F(W ) Φ C F(U) D (F(V ) D F(W )) F((U C V ) C W ) F(Φ C ) F(U C (V C W )) id D c F(U) D F(V C W ) c and the unit conditions (ii) F(1 C ) D F(U) c F(1 C C U) F(U) D F(1 C ) c F(U C 1 C ) id F(λ C ) id F(ρ C ) 1 D D F(U) λ D F(U) F(U) D 1 D ρ D F(U) A monoidal category is called strict if all the natural equivalences Φ, ρ, and λ are identities. This is the only case of interest to us in the following. In particular, it applies to the category of vector spaces with the ordinary tensor product. An additional important structure for monoidal categories is a braiding. This is an invertible functor ψ V,W : V W W V for objects V, W obeying an associativity condition. A monoidal category equipped with such a braiding is also called a braided monoidal category (or quasitensor category ). The definitions are as follows. Definition A braided monoidal category is a monoidal category C together with a

16 10 Basics natural equivalence ψ U,V : U V V U satisfying (i) id ψ U (V W ) Φ U (W V ) Φ (U W ) V (U V ) W ψ W (U V ) ψ id (W U) V Φ U (V W ) ψ (V W ) U Φ 1 (U V ) W ψ id (V U) W Φ 1 V (U W ) Φ 1 V (W U) id ψ and (ii) 1 U ψ U 1 U 1 ψ 1 U λ U ρ ρ U λ Definition A braided monoidal functor is a monoidal functor between braided monoidal categories F : C D satisfying F(U) D F(V ) F(U C V ) ψ D F(ψ C ) F(V ) D F(U) c F(V C U) Since we use braided monoidal categories extensively in the following, we introduce the customary diagrammatic notation to represent morphisms. A diagram is to be read from top to bottom. The top line and the bottom line represent tensor products of objects. The objects on the top line are connected with objects on the bottom line by strands which may cross. Over- and under-crossings are distinct and correspond to the braiding ψ and its inverse, see Figure 1.1. Strands that do not cross correspond to the identity map on that tensor factor. c

17 1.2 Representation Theory 11 V W V W W V W V ψ ψ 1 Figure 1.1: The braiding and its inverse in diagrammatic notation. = = Figure 1.2: Compatibility of braiding and tensor product. For example, we can express the hexagon identities of Definition (i) by the diagrams of Figure 1.2 (Φ is implicit). Here, crossings with close parallel strands represent a braiding with a tensor product of the corresponding objects. If ψ = ψ 1 the braiding and the category are called symmetric. It means diagrammatically that over- and under-crossings are exchangeable. The material of this section may be found in (the new edition of) Mac Lane s book on category theory [Mac98] or in some of the books on quantum groups [Maj95b, CP94] Hopf Algebra Module Categories Let A be an algebra. A left action of A on a vector space V is a linear map : A V V so that the following diagrams commute: V = k V A A V id A V id η id id V A V A V V A vector space with a left action of A is a left A-module. We denote the category of left A-modules by A M. Now, let C be a coalgebra. A left coaction of C on a vector space V is a linear map β : V C V so that the following diagrams commute: V β C V V β C V id ɛ id β id β V = k V C V id C C V

18 12 Basics A vector space with a left coaction of C is a left C-comodule. We also use the notation β(v) = v (1) v (2) for a left action, in analogy with the coproduct. Note that the index of the component living in the module is underlined. If C is a bialgebra, an element v V is called left-invariant if β(v) = 1 v. We denote the category of left C-comodules by C M. We also have the corresponding right-sided notions. We often denote left coactions by β L and right ones by β R. An important property of the categories of modules and comodules over a bialgebra is the fact that they are monoidal categories. More precisely, the tensor product of vector spaces provides a monoidal structure. Let H be a bialgebra. Let V,W be left H-modules. Then V W is a left H-module with the action a (v w) = (a (1) v) (a (2) w). (1.1) Dually, let V,W be left H-comodules. Then V W is a left H-comodule with the coaction β(v w) = v (1) w (1) (v (2) w (2) ). (1.2) This defines the monoidal structure for the categories H M and H M. The unit object in both categories is just the one-dimensional vector space isomorphic to k with trivial module (a v = ɛ(a)v) or comodule (β(v) = 1 v) structure. We can also consider more complicated types of modules. Over an algebra A, for example, we can combine left and right module structures. Let V be left A-module and a right A- module. The natural property to demand is that the module structures commute, i.e. that the diagram A V A id id V A A V V commutes. We say that V is an A-bimodule and denote the category of such objects by AM A. We can also make this category monoidal by equipping it with the tensor product over A defined by the coequalizer diagram V H W id id V W V A W (1.3) for objects V and W. The unit object is now A with the obvious bimodule structure. The action on the tensor product is given by a (v A w) = (a v) A w (1.4)

19 1.2 Representation Theory 13 and correspondingly for the right action. If H is a bialgebra, we can alternatively endow the category H M H with a monoidal structure analogous to the one of H M defined above. In this case the left action on the tensor product is given by (1.1) and the right action correspondingly. In order to distinguish between the two monoidal structures, we call the one with the tensor product and unit of Vec (the category of vector spaces) thin and the one with the unit given by H thick. We reflect this in our notation by writing a bar under thick module categories with the tensor product (1.3). We also have the dual notion of bicomodules over a coalgebra C. In the category C M C the natural monoidal structure is given by the cotensor product over C defined by the equalizer diagram V C W V W id β L β R id V C W (1.5) for objects V and W. Analogous to (1.4) the coaction on tensor products is β L (v C w) = v (1) (v (2) C w) (1.6) and accordingly for the right coaction. We denote this monoidal structure by a bar over the corresponding category. Again, for a bialgebra H we can alternatively equip H M H with the thin monoidal structure. The left coaction on a tensor product is then given by (1.2) and the right one accordingly. Let us combine two of the notions defined so far to form a category which will be important in later considerations. Consider an algebra A in the thin monoidal category H M H, i.e. an algebra so that the multiplication A A A is an H-bicomodule map (notice that the tensor product here is formally the thin one in the category and not the one of Vec). A is also called an H-bicomodule algebra. Now, consider the category of A-bimodules inside the category H M H. Equipping it with the thick tensor product (1.3) we denote this monoidal category by H A MH A. One can easily check that the kernel of the projection V W V A W (which is spanned by elements (v a) w v (a w)) is invariant under coactions of H so the tensor product A indeed exists in H A MH A and our definition is well. Note that we can also make this construction in H M leading to the monoidal category H A M A. We also have the left-right reversed notion of this and the dual ones with module and comodule structures interchanged. Over a bialgebra H we have even more possibilities. In particular, we can combine module and comodule structures. We always demand the commutativity of the various structures and denote the categories in the obvious way. Such modules are also called Hopf modules. In particular, the category H M H H is a thick monoidal category with the tensor product over H (1.3) and the category H M H H is a thick monoidal category with the cotensor product over H (1.5) (correspondingly under left-right reversal). The category H H MH H can be equipped with

20 14 Basics both tensor products. Notice for example that H H MH H is a special case of H A MH A A = H. Similar statements hold for other Hopf module categories. by setting There is another interesting type of module over a bialgebra H. Let V be a left H-module and H-comodule with the property h (1) v (1) h (2) v (2) = (h (1) v) (1) h (2) (h (1) v) (2). We say that V is a left crossed H-module (also called Yetter-Drinfeld module ). We denote the category of such objects by H HM. This category naturally has the thin monoidal structure with relations (1.1) and (1.2). We also have the corresponding right sided notion. From this point on, let H be a Hopf algebra. It turns out that the classification of the different module structures is much simplified by the following theorem. Theorem The following equivalences of categories hold. (i) M H H and Vec are equivalent, (ii) H M H H and HM are monoidal equivalent, (iii) H M H H and H M are monoidal equivalent, (iv) H H MH H, H H MH H and H HM are (pre-)braided 2 monoidal equivalent. Proof. We only sketch the construction that establishes the equivalence. For a full proof see e.g. [Sch94]. In all cases the functor establishing the equivalence between the thin and thick categories (for the purpose of this discussion we also mean the pair Vec, M H H here) is given by the assignment M M H for M an object in the thin category. M H is an object in the thick category built on the vector space M H. The right actions and coactions are given by the product and coproduct on H from the right. The left actions and/or coactions are given by (1.1) and (1.2). Conversely, for a thick module V we take its right-invariant subspace M (i.e. M = {m V β R (m) = m 1}) and find that (id S) β R projects from V to M. The projection of the left action and/or coaction gives M the structure required to be in the corresponding thin module category. On the other hand M H recovers V. For the tensor product one has (V W ) H = (V H) (W H) as required (here denotes the thin and the relevant thick tensor product). We also have the left-right reversed version of the theorem. 2 See explanation below.

21 1.3 Quantum Differentials 15 There is a thick module category that naturally has the structure of a braided category. This is H H MH H (choosing the version with the tensor product (1.3)). The braiding is given by ψ V,W (v H w) = w H v (1.7) for v V left-invariant and w W right-invariant. This determines ψ V,W completely by the requirement that it is a bimodule map (to be a morphism in the category). To be precise, this is only a pre-braiding, i.e. ψ is not necessarily invertible. However, if the antipode is a bijection, the invertibility of ψ is guaranteed. The equivalent crossed module categories are also (pre-)braided and the equivalence (iv) is an equivalence of (pre-)braided categories (see Definition 1.2.4). The (pre-)braiding in H HM is given by Correspondingly in M H H. ψ V,W (v w) = (v (1) w) v (2). The category of (co)modules of H also acquires the structure of a braided category if H carries a (co)quasitriangular structure. Let R : H H k denote a coquasitriangular structure (see Definition 1.1.2). The braiding on comodules V and W is then given by ψ V,W (v w) = R(w (1) v (1) ) w (2) v (2). (1.8) Similarly for right comodules and dually for quasitriangular structures and modules. Bibliographical Notes The treatment of Hopf modules appears e.g. in Sweedler s book [Swe69] and Abe s book [Abe80]. Crossed modules were introduced by Yetter [Yet90] and are also based on work by Drinfeld [Dri87] and Radford [Rad85]. Part (i) of Theorem is essentially the structure theorem for bimodules of Sweedler [Swe69, Theorem 4.1.1]. The further aspects of Theorem appear in the subsequent Hopf algebra literature. A complete formulation was given by Schauenburg [Sch94]. The attributes thin and thick are not standard and were introduced here for convenience. 1.3 Quantum Differentials In this section we introduce the quantum analogue of differential forms First Order Differential Calculi Definition Let A be an algebra. A first order differential calculus Ω 1 over A is an A- bimodule together with a linear map d : A Ω 1 obeying the Leibniz rule d(ab) = (da)b + a db and Ω 1 = span{a db a, b A}.

22 16 Basics This is the most general quantum version of the usual space of 1-forms over a manifold. A plays the role of the algebra of (say) smooth real valued functions on a differentiable manifold. Unfortunately, this definition appears to be far too general to be useful. To be more restrictive, we could require symmetries to be present. Classically, this would mean that a Lie group acts on the manifold, inducing an action on differential forms. This translates to a Hopf algebra H coacting on the algebra A, so that A becomes an H-comodule algebra. H plays the role of the algebra of functions on the Lie group. If A is a left H-comodule algebra we can demand that the first order differential calculus also lives in the category H M of left H-comodules instead of Vec. Similarly for right- and for bicomodules. This gives rise to the following definition. Definition In the context of Definition 1.3.1, Ω 1 is called left-/right-/bicovariant if A is a left-/right-/bicomodule over H and the actions of A on Ω 1 as well as the map d are left/right/bicomodule maps. A case of special interest is the situation where A itself is a Hopf algebra. It is then natural to demand that a first order differential calculus be bicovariant under H = A itself. This corresponds classically to the differential forms on a Lie group carrying actions by left and right translation. In the following we use differential calculus as a shorthand for bicovariant first order differential calculus. The bimodule and bicomodule structure imply that such a differential calculus lives in A A MA A. Thus, by Theorem (iv) there is a oneto-one correspondence to objects M in the category A AM of crossed modules of A. Following Woronowicz [Wor89], the further structure of a differential calculus allows to identify the objects M as submodules of A := ker ɛ A, where A is a crossed module over itself via left multiplication and left adjoint coaction: a v = av, Ad L (v) = v (1) S v (3) v (2) a A, v M. More precisely, given a crossed submodule M, the corresponding calculus is given by Γ = (A /M) A with da = π( a 1 a) (π the canonical projection). The right action and coaction on Γ are given by the right multiplication and coproduct on A, the left action and coaction by the tensor product ones with A /M as a left crossed module. Alternatively [Maj98], given in addition a Hopf algebra H dually paired with A (which we might think of as being of enveloping algebra type), we can express the coaction of A on itself as an action of H op (the Hopf algebra with the opposite product) using the pairing: h v = h, v (1) S v (3) v (2) h H op, v A. Thereby we change from the category of (left) crossed A-modules to the equivalent category of left modules of the quantum double A H op.

23 1.3 Quantum Differentials 17 In this picture the pairing between A and H descends to a pairing between A/k1 (which we may identify with A ) and H := ker ɛ H. Further quotienting A/k1 by M (viewed in A/k1) leads to a pairing with the subspace L H that annihilates M. L is called a quantum tangent space and is dual to the differential calculus Γ generated by M in the sense that Γ = Lin(L, A) via A/(k1 + M) A Lin(L, A), v a, v a (1.9) if the pairing between A/(k1 + M) and L is non-degenerate. The quantum tangent spaces are obtained directly by dualising the (left) action of the quantum double on A to a (right) action on H. Explicitly, this is the adjoint action and the coregular action h x = h (1) x S h (2), a x = x (1), a x (2) a A op, h, x H, where we have converted the right action to a left action by going from A H op -modules to H A op -modules. Quantum tangent spaces are subspaces of H invariant under the projection of this action to H via x x ɛ(x)1. Alternatively, the left action of A op can be converted to a left coaction of H which is the comultiplication (with subsequent projection onto H H ). We can use the evaluation map (1.9) to define a braided derivation on elements of the quantum tangent space via x : A A, x (a) = da(x) = x, a (1) a (2) x L, a A. This obeys the braided derivation rule x (ab) = ( x a)b + a (2) a(1) xb x L, a A. Given a right invariant basis {η i } i I of Γ with a dual basis {φ i } i I of L we have da = η i i (a) a A, i I with i := φi. (This can easily be seen to hold by evaluating against φ i i.) Exterior Differential Algebras We can further define the quantum version of a whole exterior algebra. Definition Let A be an algebra. An exterior differential algebra over A is a graded A-bimodule algebra Ω = n=0 Ωn where Ω 0 = A, together with a linear map d : Ω Ω of degree one. We require d to satisfy d 2 = 0, the graded Leibniz rule d(ab) = (da)b+( 1) a a db, and Ω n = span{a 0 da 1 da 2... da n a i A}.

24 18 Basics Notice that the span-condition implies that Ω is a quotient of the tensor algebra Ω = Ω n=0 n with Ω n = Ω 1 A Ω 1 A... A Ω 1 (n-fold). In the same way as for a first order differential calculus we define the analogous covariant versions. Definition In the context of Definition 1.3.3, Ω is called left-/right-/bicovariant if A is a left-/right-/bicomodule over H and the actions of A on Ω as well as the map d and the grading are left/right/bicomodule maps. We specialise again to the bicovariant case with A = H. It turns out that it is natural to consider exterior differential algebras with a super-hopf algebra 3 structure in this case. This comes from the following Proposition. Proposition (cf. [Brz93]). Let Γ be an H-bicovariant bimodule (that is Γ H H MH H ). The tensor algebra Γ = n=0 Γn with Γ 0 = H and Γ n = Γ H Γ H... H Γ (n-fold) is a super-hopf algebra. The coproduct and antipode are = β L + β R, S α = (S α (1) ) α (2) (S α (3) ) on degree 1 and extended to Γ as a super-hopf algebra. Proof. The proof is by induction. First note that as stated is a bimodule map since β L, β R are. We extend it by (α H β) = ( 1) α (2) β (1) (α (1) H β (1) ) (α (2) H β (2) ) which is well-defined since on α, β is a bimodule map. Moreover, for the same reason remains a bimodule map. Coassociativity on degree 1 follows from that of H and the bicomodule properties of Ω, and likewise extends to all degrees by induction. By construction, is an algebra map with the super-tensor product. Hence we have a super-bialgebra. Similarly, it is easy to see from β L, β R bimodule maps that S(h α) = (S α) S h and S(α h) = (S h) S α (S a skew-bimodule map). We extend S to higher products by S(α H β) = ( 1) α β (S β) H (S α) which is therefore well defined and remains a skew-bimodule map. That the antipode axiom is fulfilled then only has to be verified on degree 1, and extends by induction to all degrees. This is easily verified. This proposition is to be seen in connection with the previous remark that exterior differential algebras are quotients of the tensor algebra over the first order component. We 3 The definition of a super-hopf algebra is the same as for a Hopf algebra, except that it is Z 2-graded as a vector space and that the compatiblity between algebra and coalgebra structure takes the modified form (ab) = ( 1) a (2) b (1) a (1)b (1) a (2)b (2).

25 1.3 Quantum Differentials 19 can thus limit ourselves to quotients that inherit the super-hopf algebra structure of Proposition The Z 2 -grading is necessary to allow commutativity of the coproduct with d. These considerations give rise to the following definition. Definition ([Brz93]). Let H be a Hopf algebra. A bicovariant exterior differential Hopf algebra over H is a graded super-hopf algebra Ω = n=0 Ωn in H H MH H where Ω0 = H together with an H-bicomodule map d : Ω Ω of degree one. We require d to satisfy d 2 = 0, the graded Leibniz rule d(ab) = (da)b + ( 1) a a db, and to commute with coproduct and antipode. We also require Ω n = span{a 0 da 1 da 2... da n a i A}. For Hopf algebras there is a standard way of extending a bicovariant first order differential calculus to a bicovariant exterior differential algebra. This is the Woronowicz construction. It starts with the tensor algebra and quotients by an ideal obtained as the kernel of certain antisymmetriser maps. Those antisymmetrisers are similar to the classical ones, but instead of being built from permutations they are built from braidings. (For a detailed account see [Wor89].) In the case of an ordinary Lie group the construction reduces to the ordinary construction of the exterior algebra out of the space of 1-forms. Bibliographical Notes For differential calculi on quantum groups see Woronowicz s seminal paper [Wor89]. A textbook treatment of some of the material can be found, e.g., in [KS97].

26 Chapter 2 Twisting Theory Although there does not as yet seem to be a satisfactory concept of Hopf algebra cohomology, certain aspects of it are known. In particular, 1- and 2-cochains and corresponding boundary operators can be defined. These generalise the respective notions of both Lie algebra and group cohomology. However, Hopf algebra cohomology is much richer due to its noncommutativity and more symmetrical due to the self-duality of the Hopf algebra axioms. Remarkably, a 2-cocycle (i.e. a closed 2-cochain) on a Hopf algebra gives rise to a deformation, called a twist. This is a purely noncommutative effect which disappears upon restriction to groups or Lie algebras. It turns out to be related to deformation quantisation as will be briefly reviewed in Section 2.3. In this chapter we study the influence of the twist deformation on the representation theory. In the simplest case, if two Hopf algebras are related by a (coproduct) twist, their respective categories of modules are equivalent. This was found by Drinfeld who introduced the concept of twisting [Dri90]. We extend this result in Section 2.2 by showing that such an equivalence holds in fact for many types of representation categories (as introduced in Section 1.2.2). We then proceed to apply the results to quantum differential geometry. Namely, we show in Section 2.4 that they give rise to a one-to-one correspondence between quantum differential calculi over quantum spaces that are related by a (product) twist. In particular, this means that a cocycle deformation quantisation (see Section 2.3) of an ordinary group naturally carries a deformation quantised differential calculus. This will be of interest in Chapter 4, where such a quantum group is studied as a toy model for Planck scale physics. 20

27 2.1 Hopf Algebras Hopf Algebras We review in this section the theory of twists on Hopf algebras. This provides a way of obtaining new Hopf algebras from given ones using elements of a Hopf algebra cohomology theory. Twists were introduced by Drinfeld [Dri90] in the context of quasi-hopf algebras. See also [Maj95b, CP94]. Let H denote a Hopf algebra. Definition An element χ H H is a counital 2-cocycle if it has the following properties. (i) χ is invertible, i.e. there exists χ 1 H H so that χχ 1 = χ 1 χ = 1 1. (ii) (1 χ)(id )χ = (χ 1)( id)χ. (cocycle condition) (iii) (id ɛ)χ = (ɛ id)χ = 1. (counitality) Proposition A counital 2-cocycle χ H H defines a twisted Hopf algebra H χ with the same algebra structure and counit as H. The coproduct and antipode are given by χ h = χ( h)χ 1, S χ h = U(S h)u 1 with U = χ (1) S χ (2). If H has a quasitriangular structure R H H, then H χ has a quasitriangular structure R χ given by R χ = χ 21 Rχ 1. Proof. Sketch (for the Hopf algebra structure): The associativity of the twisted coproduct follows from the cocycle condition (ii) together with the corresponding condition for χ 1 using (i). That the counit remains a counit follows from the unitality (iii) with (i). That the twisted coproduct remains an algebra map is obvious from its definition. It remains to check the antipode property for S χ, which is done by explicit calculation. For a complete proof see e.g. [Maj95b]. Note that the inverse operation to twisting with χ is twisting with χ 1. In particular, χ 1 is a counital 2-cocycle with respect to H χ. By dualising this setting we obtain twists that modify the product structure instead of the coproduct structure. Definition A linear map χ : H H k is a unital 2-cocycle if it has the following properties. (i) χ is convolution invertible, i.e. there exists χ 1 : H H k so that χ(a (1) b (1) )χ 1 (a (2) b (2) ) = χ 1 (a (1) b (1) )χ(a (2) b (2) ) = ɛ(a) ɛ(b) a, b H.

28 22 Twisting Theory (ii) χ(a (1) b (1) )χ(a (2) b (2) c) = χ(b (1) c (1) )χ(a b (2) c (2) ) a, b, c H. (cocycle condition) (iii) χ(a 1) = χ(1 a) = ɛ(a) a H. (unitality) Proposition A unital 2-cocycle χ : H H k defines a twisted Hopf algebra H χ with the same coalgebra structure and unit as H. The product and antipode are given by a b = χ(a (1) b (1) ) a (2) b (2) χ 1 (a (3) b (3) ), S χ a = U(a (1) ) S a (2) U 1 (a (3) ) with U(a) = χ(a (1) S a (2) ). If H has a coquasitriangular structure R : H H k, then H χ has a coquasitriangular structure R χ given by R χ (a b) = χ(b (1) a (1) ) R(a (2) b (2) ) χ 1 (a (3) b (3) ). (2.1) Proof. Sketch (for the Hopf algebra structure): The associativity of the twisted product follows from the cocycle condition (ii) together with the corresponding condition for χ 1 using (i). That the unit remains a unit follows from the unitality (iii) with (i). That the twisted product remains a coalgebra map is obvious from its definition. It remains to check the antipode property for S χ, which is done by explicit calculation. For a complete proof see e.g. [Maj95b]. Again, χ 1 is a unital 2-cocycle with respect to H χ and defines the inverse twist. 2.2 Module Categories Drinfeld showed [Dri90] that a twist of a (quasitriangular) Hopf algebra extends to its category of modules. More precisely, it gives rise to an equivalence between the (braided) monoidal categories of modules of the original and the twisted Hopf algebra. (See Theorem below, in dual formulation.) We show that similar equivalences hold for other kinds of module categories of Hopf algebras as well. Since this case is more relevant in the following, we limit ourselves to the product twists. Each statement has an obvious dual version with coproduct twist. Theorems and are due to joint work with Shahn Majid [MO99]. To simplify the notation, we denote actions as multiplications. In particular, twisted actions are denoted with a. For clarity, we denote here (in contrast to Section 1.2.2) a thin tensor product by and its twisted counterpart by χ.